Finite Group Databasebeta

← #29697318

Subgroups — #29697318

Order 15616.

1010 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839761C211yesmaximal, characteristic
2#21448659C211yesmaximal, characteristic
3#21446515C211yesmaximal, characteristic, complemented
4#21446501C211yesmaximal, characteristic, complemented
5#21446446C211yesmaximal, characteristic, complemented
6#21446439C211yesmaximal, characteristic, complemented
7#21446086C211yesmaximal, characteristic, complemented
8#21446072C211yesmaximal, characteristic, complemented
9#21445909C211yesmaximal, characteristic, complemented
10#21445850C211yesmaximal, characteristic, complemented
11#21445757C211yesmaximal, characteristic
12#21445652C211yesmaximal, characteristic, complemented
13C4×#14519700C211yesmaximal, characteristic, complemented
14#21445603C211yesmaximal, characteristic, complemented
15C61×#2344C211yesmaximal, characteristic, complemented
16C22×#11758222C2211yescharacteristic
17C22×#11758194C2211yescharacteristic
18C2×C4×Dic122C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20C2×#14519710C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22#16839761C418yescomplemented
23C2×#14519700C2211yescharacteristic
24C2×#14519700C2211yescharacteristic
25C2×#14519700C2211yescharacteristic
26C2×#14519700C2211yescharacteristic
27C2×#14519697C2211yescharacteristic
28C2×#14519697C2211yescharacteristic
29C2×#14519697C2211yescharacteristic
30C2×#14519697C2211yescharacteristic
31C2×C22⋊#92418C2211yescharacteristic
32C2×#14519688C2211yescharacteristic
33C2×C4×#92418C2211yescharacteristic
34#16839664C2211yescharacteristic
35#16839664C2211yescharacteristic
36#16839662—21no—
37#16839662—21no—
38#16839622C2211yescharacteristic
39#16839621—21no—
40#16839609—21no—
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839517C2211yescharacteristic
48#16839517C2211yescharacteristic
49C2×#117×C61—21no—
50C22×C4⋊C4×C61C2211yescharacteristic
51C61×#389C2211yescharacteristic
52C61×#389C2211yescharacteristic
53C61×#387—21no—
54C61×#387—21no—
55C61×#381—21no—
56C61×#380C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C4×C4⋊C4×C61C2211yescharacteristic
59C4×C22⋊C4×C61C2211yescharacteristic
60C2×#11758222—22no—
61C2×#11758222—22no—
62C2×#11758222—22no—
63C2×#11758222—41no—
64C2×#11758222—41no—
65C2×#11758222C2×C414yes—
66C22×Dic122C2311yescharacteristic
67C22×D244C2311yescharacteristic
68C22×C22⋊D61—21no—
69C22×C4×D61—21no—
70C2×#11758194—41no—
71C2×#11758194—41no—
72C2×#11758194C2×C414yes—
73C22×#92418C2311yescharacteristic
74C22×#92418C2311yescharacteristic
75C22×#92418C2311yescharacteristic
76C22×#92418C2311yescharacteristic
77C4×Dic122—22no—
78C4×Dic122—22no—
79C4×Dic122—41no—
80C4×Dic122—41no—
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×C22⋊Dic61—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61—21no—
92#14519710—41no—
93#14519708—41no—
94#14519700—22no—
95#14519700—22no—
96#14519700C2×C414yes—
97#14519700C2×C414yes—
98#14519697—41no—
99#14519697—41no—
100#14519697C2×C414yes—
101#14519697C2×C414yes—
102C22⋊#92418—22no—
103#14519688—41no—
104C4×#92418—41no—
105#14519684—21no—
106C23×C4×C61—21no—
107C23×C4×C61C2311yescharacteristic
108#117×C61—44no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61—22no—
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61—21no—
113C2×C4⋊C4×C61C2×C414yes—
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C22⋊C4×C61—21no—
119C2×C22⋊C4×C61D411yescharacteristic
120C2×C22⋊C4×C61D411yescharacteristic
121C2×C42×C61—21no—
122C2×C42×C61—21no—
123C2×C42×C61C2311yescharacteristic
124C2×C42×C61C2311yescharacteristic
125#95×C61—21no—
126#95×C61—21no—
127#95×C61—21no—
128#95×C61—21no—
129#95×C61—21no—
130#95×C61D411yescharacteristic
131#95×C61D411yescharacteristic
132C24×C61C2×D411yescharacteristic
133C23×D61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—24no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—42no—
144C22×C4×C61—42no—
145C22×C4×C61C22×C412yes—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61#4112yes—
152C22×C4×C61C2411yescharacteristic
153C2×C22⋊D61—24no—
154C2×C22⋊D61—42no—
155C2×C22⋊D61—42no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159#11758222—28no—
160#11758222—44no—
161#11758222—44no—
162#11758222—44no—
163C2×D244—21no—
164C2×D244—21no—
165C2×D244—21no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244C22×C412yes—
169C2×C4×D61—24no—
170C2×C4×D61—42no—
171C2×C4×D61—42no—
172C2×Dic122—21no—
173C2×Dic122—21no—
174C2×Dic122—21no—
175C2×Dic122C22×C412yes—
176C2×Dic122#4112yes—
177C2×Dic122#4112yes—
178C4⋊C4×C61—24no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—42no—
186C4⋊C4×C61—42no—
187C4⋊C4×C61C22×C412yes—
188C4⋊C4×C61C22×C412yes—
189C22⋊C4×C61—22no—
190C22⋊C4×C61—41no—
191C22⋊C4×C61—42no—
192C42×C61—21no—
193C42×C61—21no—
194C42×C61—21no—
195C42×C61—21no—
196C42×C61—42no—
197C42×C61—42no—
198C22⋊Dic61—81no—
199#11758198—42no—
200#11758198—42no—
201#11758198—81no—
202#11758198—81no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207#11758196—42no—
208#11758196—42no—
209C4×Dic61—42no—
210C4×Dic61—42no—
211#11758194—44no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418C22×C412yes—
223C2×#92418C22×C412yes—
224C23×C61—21no—
225C23×C61—21no—
226C23×C61—21no—
227C23×C61—21no—
228C23×C61—21no—
229C23×C61—21no—
230C23×C61C4×D412yes—
231C23×C61C22×D411yescharacteristic
232C22×D61—22no—
233C22×D61—41no—
234C22×D61—41no—
235C22×D61—41no—
236C22×D61—41no—
237C22×D61—41no—
238C22×D61—41no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61—22no—
247C2×C4×C61—22no—
248C2×C4×C61—22no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—21no—
256C2×C4×C61—21no—
257C2×C4×C61—21no—
258C2×C4×C61—48no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61—41no—
265C2×C4×C61—41no—
266C2×C4×C61—41no—
267C2×C4×C61—41no—
268C2×C4×C61—41no—
269C2×C4×C61C4×D412yes—
270C2×C4×C61AGL(1,ℤ/8)12yes—
271C2×C4×C61C23×C411yescharacteristic
272C2×C4×C61C2×#4111yescharacteristic
273C2×C4×C61C2×#4111yescharacteristic
274C2×C4×C61C2×#4111yescharacteristic
275C2×C4×C61H(5,𝔽2)11yescharacteristic
276C2×C4×C61H(5,𝔽2)11yescharacteristic
277C22⋊D61—48no—
278C2×Dic61—22no—
279C2×Dic61—22no—
280C2×Dic61—22no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287D244—22no—
288D244—22no—
289D244—41no—
290D244—41no—
291C4×D61—48no—
292Dic122—22no—
293Dic122—22no—
294Dic122—22no—
295Dic122#11714yescomplemented
296#92418—22no—
297#92418—22no—
298#92418—41no—
299#92418—41no—
300#22583—611nomaximal, Sylow
301C22×C61—22no—
302C22×C61—22no—
303C22×C61—22no—
304C22×C61—41no—
305C22×C61—41no—
306C22×C61—41no—
307C22×C61—41no—
308C22×C61C2×C4×D411yescharacteristic
309C22×C61#54611yescharacteristic
310C22×C61#54611yescharacteristic
311C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
312C22×C61#57511yescharacteristic
313C22×C61#57711yescharacteristic
314C22×C61#57811yescharacteristic
315C2×D61—42no—
316C2×D61—42no—
317C2×D61—42no—
318C2×D61—81no—
319C2×D61—81no—
320C2×D61—81no—
321C2×D61—81no—
322C4×C61—22no—
323C4×C61—22no—
324C4×C61—41no—
325C4×C61—42no—
326C4×C61—42no—
327C4×C61—41no—
328C4×C61—41no—
329C4×C61—81no—
330C4×C61—81no—
331C4×C61C2×#11711yescharacteristic
332C4×C61#51811yescharacteristic
333C4×C61#55111yescharacteristic
334C4×C61#55111yescharacteristic
335Dic61—24no—
336Dic61—81no—
337Dic61—81no—
338C2×#417—611no—
339#2344D6111yescharacteristic, complemented
340#2341—611no—
341#1936—611no—
342#1934—611no—
343#1926—611no—
344#1925—611no—
345#1865—611no—
346#1863—611no—
347#1824—611no—
348#1809—611no—
349#1788—611no—
350#1763—611no—
351C4×#103—611no—
352#1753—611no—
353C2×C61—42no—
354C2×C61#234211yescharacteristic
355C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
356C2×C61#294711yescharacteristic
357C2×C61#330811yescharacteristic
358C2×C61#331711yescharacteristic
359C2×C61#332711yescharacteristic
360C2×C61#333211yescharacteristic
361D61—82no—
362C22×#41—611no—
363C22×M16—611no—
364C2×C4×Q8—611no—
365C2×C4×Q8—611no—
366C2×#117—21no—
367C22×C4⋊C4C2×D6111yescharacteristic
368C2×#107—611no—
369C2×#106—611no—
370#417—618no—
371C2×#103—611no—
372C2×#103—611no—
373C2×#103—611no—
374C2×#103—611no—
375C2×#102—611no—
376C2×#102—611no—
377C2×#102—611no—
378C2×#102—611no—
379C2×C22⋊C8—611no—
380C2×#97—611no—
381C2×C4×C8—611no—
382#390—611no—
383#390—611no—
384#390—1221no—
385#390—1221no—
386#389C2×D6111yescharacteristic
387#389C2×D6111yescharacteristic
388#387—21no—
389#387—21no—
390#381—21no—
391#380—611no—
392#380—1221no—
393#380C2×D6111yescharacteristic
394C4×C4⋊C4C2×D6111yescharacteristic
395C4×C4⋊C4C2×D6111yescharacteristic
396C4×C22⋊C4—1221no—
397C4×C22⋊C4C2×D6111yescharacteristic
398#340—611no—
399#340—611no—
400#340—611no—
401#340—611no—
402#340—611no—
403#340—611no—
404#336—611no—
405#336—611no—
406C61#2258311yescharacteristic, complemented, Sylow
407C2×#41—614no—
408C2×#41—1222no—
409C2×#41—1222no—
410C2×#41—1222no—
411C2×#41—2441no—
412C2×#41—2441no—
413C22×Q8—611no—
414C22×D4—611no—
415C22×D4—1221no—
416C23×C4—21no—
417C23×C4—1221no—
418C23×C4C22×D6111yescharacteristic
419C2×M16—614no—
420C2×M16—2441no—
421C2×M16—2441no—
422C22×C8—611no—
423C22×C8—611no—
424C22×C8—611no—
425C22×C8—611no—
426C4×Q8—1222no—
427C4×Q8—1222no—
428C4×Q8—2441no—
429C4×Q8—2441no—
430#117—44no—
431C2×C4⋊C4—22no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—21no—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—611no—
436C2×C4⋊C4—611no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4C4×D6114yes—
440C2×C4⋊C4C22×D6111yescharacteristic
441C2×C4⋊C4C22×D6111yescharacteristic
442C2×C4⋊C4C22×D6111yescharacteristic
443C2×C4⋊C4C22×D6111yescharacteristic
444C2×C22⋊C4—21no—
445C2×C22⋊C4—1221no—
446C2×C22⋊C4—1221no—
447C2×C22⋊C4—1221no—
448C2×C22⋊C4—1221no—
449C2×C22⋊C4—1221no—
450C2×C22⋊C4C22⋊D6111yescharacteristic
451C2×C22⋊C4C22⋊D6111yescharacteristic
452C2×C42—21no—
453C2×C42—21no—
454C2×C42—1221no—
455C2×C42—1221no—
456C2×C42C22×D6111yescharacteristic
457C2×C42C22×D6111yescharacteristic
458#107—2441no—
459#106—2441no—
460#103—614no—
461#103—614no—
462#103—1222no—
463#103—1222no—
464#102—614no—
465#102—614no—
466#102—2441no—
467#102—2441no—
468C22⋊C8—1222no—
469#97—2441no—
470C4×C8—2441no—
471#95—21no—
472#95—21no—
473#95—21no—
474#95—21no—
475#95—21no—
476#95—1221no—
477#95C22⋊D6111yescharacteristic
478#95C22⋊D6111yescharacteristic
479C24—1221no—
480C24C2×C22⋊D6111yescharacteristic
481#41—1228no—
482#41—2444no—
483#41—2444no—
484#41—2444no—
485C2×Q8—612no—
486C2×Q8—612no—
487C2×Q8—612no—
488C2×Q8—1221no—
489C2×Q8—1221no—
490C2×Q8—1221no—
491C2×D4—612no—
492C2×D4—1224no—
493C2×D4—1221no—
494C2×D4—1221no—
495C2×D4—1221no—
496C2×D4—1221no—
497C2×D4—1221no—
498C2×D4—2442no—
499C2×D4—2442no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—24no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—42no—
510C22×C4—42no—
511C22×C4—1221no—
512C22×C4—1224no—
513C22×C4—1221no—
514C22×C4—1221no—
515C22×C4—2442no—
516C22×C4—2442no—
517C22×C4C2×C4×D6112yes—
518C22×C4#1175822212yes—
519C22×C4C2×C22⋊D6111yescharacteristic
520C22×C4C2×C22⋊D6111yescharacteristic
521C22×C4C2×C22⋊D6111yescharacteristic
522C22×C4C2×C22⋊D6111yescharacteristic
523C22×C4C2×C22⋊D6111yescharacteristic
524C22×C4C23×D6111yescharacteristic
525M16—2444no—
526C2×C8—612no—
527C2×C8—612no—
528C2×C8—1221no—
529C2×C8—1221no—
530C2×C8—1221no—
531C2×C8—1221no—
532C2×C8—1221no—
533C2×C8—1221no—
534C2×C8—1221no—
535C2×C8—1221no—
536C2×C8—1221no—
537C2×C8—1221no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—24no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—1221no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—1221no—
551C4⋊C4—2442no—
552C4⋊C4—2442no—
553C4⋊C4C2×C4×D6112yes—
554C4⋊C4C2×C4×D6112yes—
555C22⋊C4—22no—
556C22⋊C4—41no—
557C22⋊C4—42no—
558C22⋊C4—4881no—
559C22⋊C4—4881no—
560C22⋊C4—4881no—
561C22⋊C4—2442no—
562C22⋊C4—2442no—
563C42—21no—
564C42—21no—
565C42—21no—
566C42—21no—
567C42—42no—
568C42—42no—
569C42—2442no—
570C42—2442no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—1222no—
578C23—2441no—
579C23—2441no—
580C23—2441no—
581C23—2441no—
582C23—2441no—
583C23—2441no—
584C23C4×C22⋊D6112yes—
585C23C22×C22⋊D6111yescharacteristic
586Q8—614no—
587Q8—1222no—
588Q8—1222no—
589Q8—1222no—
590D4—1222no—
591D4—1222no—
592D4—2448no—
593D4—2441no—
594D4—2441no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—48no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—1222no—
626C2×C4—1222no—
627C2×C4—1222no—
628C2×C4—2448no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4—2441no—
634C2×C4—2441no—
635C2×C4C4×C22⋊D6112yes—
636C2×C4#1451980312yes—
637C2×C4C22×C4×D6111yescharacteristic
638C2×C4#41×D6111yescharacteristic
639C2×C4#41×D6111yescharacteristic
640C2×C4C2×#1175822211yescharacteristic
641C2×C4#1451982311yescharacteristic
642C2×C4#1451982311yescharacteristic
643C8—1222no—
644C8—1222no—
645C8—2441no—
646C8—2441no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22—4881no—
655C22—4881no—
656C22—4881no—
657C22—4881no—
658C22—2442no—
659C22—2442no—
660C22—2442no—
661C22C2×C4×C22⋊D6111yescharacteristic
662C22#1684040011yescharacteristic
663C22#1684040011yescharacteristic
664C22C2×#1451980311yescharacteristic
665C22#1684056011yescharacteristic
666C22#1684057011yescharacteristic
667C22#1684057711yescharacteristic
668C4—22no—
669C4—22no—
670C4—41no—
671C4—42no—
672C4—42no—
673C4—41no—
674C4—41no—
675C4—81no—
676C4—81no—
677C4—1224no—
678C4—4881no—
679C4—4881no—
680C4#117×D6111yescharacteristic
681C4#1684023611yescharacteristic
682C4#1684043611yescharacteristic
683C4#1684043611yescharacteristic
684C2—42no—
685C2—4882no—
686C2#2144866911yescharacteristic
687C2C4×#1451980311yescharacteristic
688C2#2145466211yescharacteristic
689C2#2145780411yescharacteristic
690C2#2145793911yescharacteristic
691C2#2145808111yescharacteristic
692C2#2145814011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2344C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenter#2344D61
hypocenterC61#22583
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2344

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #21446515C2semidirect
4. #21446501C2semidirect
5. #21446446C2semidirect
6. #21446439C2semidirect
7. #21446086C2semidirect
8. #21446072C2semidirect
9. #21445909C2semidirect
10. #21445850C2semidirect
12. #21445652C2semidirect
13. C4×#14519700C2semidirect
14. #21445603C2semidirect
15. C61×#2344C2semidirect
22. #16839761C4semidirect
295. Dic122#117semidirect
339. #2344D61semidirect
406. C61#22583semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168397612
#214486592
#214465152
#214465012
#214464462
#214464392
#214460862
#214460722
#214459092
#214458502
#214457572
#214456522
C4×#145197002
#214456032
C61×#23442
#2258361